Carbon-Based Metal Free Catalysts: Preparation, Structural and Morphological Property and Application

Let V be a d-dimensional vector space over finite field F equipped with non-degenerate hermitian, alternating, or quadratic form. Suppose |F|=q2 if is and |F|=q otherwise. Given integers e,e′ such that e+e′⩽d, we estimate the proportion of pairs (U,U′), where U e-subspace U′ e′-subspace V, U∩U′=0 U⊕U′ (the sum direct usually not perpendicular). The shown to positive at least 1−c/q>0 for some constant c. For example, c=7/4 suffices in both unitary symplectic cases. arguments orthogonal case are delicate assume dim⁡(U) dim⁡(U′) even, an assumption relevant algorithmic application (which discuss) recognising classical groups. We also describe how groups G relies on connection between certain (U,U′) subspaces (g,g′)∈G2 group elements U=im(g−1) U′=im(g′−1).